Entanglement enhancement and postselection for two atoms interacting with thermal light K. Bra´dler and R. Jáuregui Instituto de F´ısica, Apdo. Postal 20-364, México 01000, México∗ (Dated: February 1, 2008) The evolution of entanglement for two identical two-level atoms coupled to a resonant thermal field is studied for two different families of input states. Entanglement enhancement is predicted 7 fora well definedregion of theparameter spaceof oneof these families. Themost intriguingresult 0 isthepossibility ofprobabilistic production ofmaximally entangled atomic states evenif theinput 0 atomic state is factorized and the corresponding output state is separable. 2 n PACSnumbers: 42.50.Dv,03.67.-a a Keywords: Atom-lightinteraction,Entanglement generation J 2 2 Entanglement of atomic systems is a promising resourcefor performing various computational and communication tasks originated in quantum information theory. Recent experimental progress has been achieved in the field of 3 creation of multipartite highly nonclassical states in a linear Paul trap [1, 2]. Production of such states requires a v nontrivial implementation of a sequence of nonlocal operations (unitary gates) that entangle originally uncorrelated 6 atoms in a desired way. Apart from these sophisticated methods for manipulating the atomic states with the help of 4 a precisely tuned laser field, it has been predicted that originallyindependent atoms may become entangledafter the 1 6 interactionwithathermalelectromagneticfield. Atfirstsightthismayseemcounterintuitivebecausethermallightis 0 considered to be a decoherence source. But indeed, if we stay in the quite modest case of entangling just two atoms, 6 Kim et al. [3] as well as Bose et al. [4] and An et al. [5] have reported the occurrence of entanglement of initially 0 separable mixed states after the interaction with thermal light. These studies belong to a broader group of works on h/ the behavior of an atomic ensemble coupled to different kinds of bosonic environment [6, 7]. p Loosely speaking, all previous studies of the coupling of an atomic system to a thermal bath [3, 4, 5, 8] concluded - with the interesting result that entanglement not only does not disappear after the interaction but it can be even t n created from originally separable states. In this work we go further. In the framework of the model describing a the interaction of an atomic ensemble and a single mode photon field worked out by Tavis et al. [9] we study two u physically relevant families of initial pure atomic states coupled to a thermal bath. The states can be tuned by q changingits Schmidtnumber (characterizingthe degree ofentanglementofinitial states)fromfactorizedstates to all : v four maximally entangled states (Bell states). The calculations give quite unexpected results that, to our knowledge, i have not been reported so far. It is found that for an input state of the form Ψ = sinβ 00 +cosβ 11 and X β | i | i | i for a sufficiently ’hot’ environment, there is a stable nonzero probability of producing a maximally entangled state r a irrespective of the value of β and, what is more interesting, despite of the fact that the resulting state after the interaction is separable. This procedure is usually called postselection. Thus we present here a probabilistic source of maximally entangled atomic pairs. Adetailedstudyofthefamilyofinputatomicstates Φ =sinβ 01 +cosβ 10 showsthatentanglementofoutput β | i | i | i states is higher in comparison with input states for a wide interval of β values. The picture started in Refs. [3, 6] is now completed by showing that the entangling propertiesof a thermal field prevailevenat verylow temperatures, so that vacuum fluctuations can generate partially entangled states from initially factorized states. The structure of the paper is as follows. In Section I we introduce the entanglement of formation(EOF) [10]. This entanglement measure is calculated for general two-qubit states with the help of the analytical expression for the concurrence [11]. In Section II we describe the Tavis-Jaynes-Cummings model setting the stage for the main results presentedin two subsections where the evolutionof thermal lightwith two different atomic input configurations Φ β | i and Ψ is studied. The details about the calculations are given in Appendix. β | i ∗Electronicaddress: kbradler@epot.cz, rocio@fisica.unam.mx 2 I. ENTANGLEMENT MEASURES Mostaxiomsrequiredforentanglementmeasures[12]areaccomplishedbytheentanglementofformation[10](EOF) defined by E (̺)= inf p E(ϕ ϕ ), (1) F i i ̺=Pipi|ϕihϕ|i i | ih | X where E( ) = S(Tr [ϕ ϕ ]) is called the entropy of entanglement (the von Neumann entropy of the state of 1(2) i • | ih | interest that is traced over one of its subsystems). The EOF is generally uneasy to calculate even for the lowest dimensionalsystems. However,Hill andWootters proved[11]thatthe EOFcanbe calculatedanalyticallyinthe case of an arbitrary two qubit state ̺. They showed that 1 E (̺)=h 1+ 1 C2(̺) , (2) F 2 − (cid:18) (cid:16) p (cid:17)(cid:19) where h(x)= xlog (x) (1 x)log (1 x) is the Shannon entropy and − 2 − − 2 − C(̺)=max 0,λ λ λ λ . (3) 1 2 3 4 { − − − } The parameters λ in Eq. (3) are eigenvalues in descending order of the square root of the matrix i R˜(̺)=̺(σ σ )̺ (σ σ ) (4) y y ∗ y y ⊗ ⊗ with representing the complex conjugation in the standard basis and σ is the y-Pauli matrix. C is called the y ∗ concurrence and since the EOF is monotonous in C, the concurrence can also be considered as the entanglement measure. Nevertheless, as emphasized in [13], the proper entanglement measure is the EOF. Using the concurrence forthispurposecouldcauseconfusionwheninvestigatingentanglementofsystemswithdimensionalityn nforn=2 × 6 since in that case the definition of concurrence is not unique. Also in some cases, e.g. in [3], the negativity [14] as an entanglement measure is preferred. The reasonis that it is easier to calculate (tracing over a partially transposed density matrix). On the other hand, the negativity does not coincide with the entropy of entanglement introduced above for pure bipartite states where it is the unique measure of entanglement [15]. Rather recently [16], connection of the negativity with the concurrence (and thus to the EOF that coincides with the entropy of entanglement for pure bipartite states) was discovered in terms of convex roof construction. Anyhow, in this paper the particular forms of the density matrices allow us to calculate the EOF directly from the concurrence. The second note concerns the term Schmidt number. It characterizes the degree of entanglement of input states but does not uniquely determine the behavior after the interaction. In other words,even if two input states have the same Schmidt number, for example two Bell states, their evolution and thus entanglement after the interaction may be completely different. II. INTERACTION OF TWO ATOMS WITH THERMAL LIGHT The Hamiltonianoftwoidenticaltwo-levelatoms interactingwitha single-modeelectromagneticfieldin the dipole approximation and standard notation is given by H = H +H 0 int 1 ~Ω 2 = ~Ω(cid:18)b†b+ 2(cid:19)+ 2 (cid:16)σz(1)+σz(2)(cid:17)+~g Xi=1σ+(i)b+σ−(i)b†!, (5) whereresonanceofthe photonenergyandtheatomiclevelsplittingisassumed,andtherotating-waveapproximation is used. The analogous Hamiltonian for N-atoms can be analytically diagonalized as already shown in Ref. [9]. Here we use the dressed states formalism to obtain the eigenvalues and eigenvectors. Since [H ,H ] = 0, H induces 0 int int transitionsonlybetweenthedegeneratestatesofH that,foragivenfieldexcitationnumbern,constitutesthetetrad 0 n 11 , n+1 10 , n+1 01 , n+2 00 . The bare states form a ’semilogical’ basis where, e.g., n+1 10 n {| i| i | i| i | i| i | i| i} | i| i represents the state with (n+1) photons, the first atom in the excited state (logical state one) and the second one in the ground state (logical state zero). Expressing the interaction Hamiltonian in this basis we get a block diagonal 3 matrix with the n-th block given by 0 √n+1 √n+1 0 H(n) =~g√n+1 0 0 √n+2. (6) int √n+1 0 0 √n+2  0 √n+2 √n+2 0      After normalization of the eigenvectors of this matrix, the dressed basis states read n+1 n+2 1 = n 11 + n+2 00 (7a) | in r2n+3"−rn+1| i| i | i| i# 1 2 = [n+1 10 n+1 01 ] (7b) | in 2 | i| i−| i| i r 2 n+2 2n+3 2n+3 3 = n 11 n+1 10 n+1 01 + n+2 00 (7c) | in r4n+6"rn+1| i| i−r2n+4| i| i−r2n+4| i| i | i| i# n+2 n+2 2n+3 2n+3 4 = n 11 + n+1 10 + n+1 01 + n+2 00 . (7d) | in r4n+6"rn+1| i| i r2n+4| i| i r2n+4| i| i | i| i# In particular, the atomic state 1/√2(10 01 ) is a dark state. | i−| i A. Initial Atomic State Φβ | i In this section, the system light-atoms is assumed to be initially in a factorized state ̺(Φ)(β,0) = ̺ ̺(Φ) light ⊗ atoms where the atomic state is a pure state Φ =sinβ 01 +cosβ 10 (8) β | i | i | i with the Schmidt number sinβ and the light considered to be in a thermal state ̺ = 1 e ~Ω/kT e~Ω/2kT ∞ e (n+1/2)~Ω/kT n n light − − − | ih | h i nX=0 ∞ en¯n¯n = n n, (9) n! | ih | n=0 X with n¯ = e~Ω/kT 1 −1 the average number of photons. After a direct but otherwise tedious calculation the time − evolveddensity matrix ̺(Φ)(β,t) can be obtained. Since we are interested in the entanglement behavior of the atom- (cid:2) (cid:3) atom system, we trace over the electromagnetic field states. The resulting time-dependent reduced density matrix is of the form a 0 0 0 1 0 a a 0 ̺˜(Φ)(β,t)= 2 3 . (10) 0 a a 0 3 4 0 0 0 a  5   The specific expressions of the matrix elements can be found in Appendix. Thestructureof̺˜(Φ) isveryfriendlyforthecalculationofthe concurrencewithEq.(3)andconsequentlythe EOF. The result is illustrated in Fig. 1 where oscillations of the EOF are noticeable for all β. There is a β 1 that 0 ≪ depends on n¯ such that for the subinterval β < β < π/2 β the minimum value of the EOF is zero and the local 0 0 − maxima (revivals) are always smaller than the initial value for n¯ > 0 and equal for n¯ = 0. Thus, the EOF of the atomic entangledstates is reducedin the presence ofthermallight. As the temperature increasesthe temporal width of the revivals decreases. A brief inspection of the picture reveals other interesting aspects of interacting light with this family of states Φ in the second interval of β, π/2 < β < π . Due to the fact that 1/√2(10 01 ) is a β | i | i−| i dark state the behavior is completely different. The EOF exhibits very soft oscillations and is never smaller than its initial value. We thus see that in this region, the resulting state is stable under the interaction with thermal light. This could be an important hint for quantum engineers (and protocol designers generally) telling them that there 4 1 1 0.8 0.8 0.6 0.6 F F O O E0.4 E0.4 0.2 0.2 0 0 10 10 8 8 6 0 6 0 0.39 0.39 0.79 0.79 gt 4 1.18 gt 4 1.18 1.57 1.57 2 1.96 2 1.96 2.752.36 β [rad] 2.752.36 β [rad] 3.14 3.14 (a) (b) FIG. 1: The EOF of the output state given by Eq. (10) for interacting atomic states Φβ with: (a) a zero photon field, and | i (b) a thermal field with a nonzero average photon number (n¯ 0.64) . A completely different behavior for 0 β π/2 and ≈ ≤ ≤ π/2 β π can be seen. EOF revivals are observed in the first interval where, generally, entanglement of input states is ≤ ≤ devaluedexceptforinitially factorized orweakly entangledstates. Ontheotherhand,inthesecond intervaltheEOFisnever lower than its initial value. Wecan observe significant values of the EOFfor initially factorized atomic states and n¯ =0. exists an important class of initial atomic states where the interaction with a thermal field means not only no loss of entanglement but even its enhancement without strong fluctuations. The works presented so far on this topic (e.g. Refs. [3, 4]) were limited only to input atomic states with small values of the Schmidt number 0 < β < β 1 and 0 ≪ recognized the presence of the dark state but do not discuss the stability of entanglement in the whole range of β. For n¯ =0 the matrix elements of Eq. (10) have a particulary simple form 1 a = (1+sin2β)(1 cos2α t) (11a) 1 1 4 − − 1 1 a = (1+sin2β)(1+cos2α t) cos2βcosα t+ (1 sin2β) (11b) 2 1 1 8 − − − 4 − 1 1 a = (1+sin2β)(1+cos2α t) (1 sin2β) (11c) 3 1 8 − − 4 − 1 1 a = (1+sin2β)(1+cos2α t)+cos2βcosα t+ (1 sin2β) (11d) 4 1 1 8 − − 4 − a = 0. (11e) 5 Byexaminingthedensitymatrixforβ =0wemayexactlyseetheinfluenceofthevacuumfluctuationsonafactorized input state. We see that there is a 50% probability of finding the system with both atoms in their ground state and a single photon, while there is a 50% probability that this photon has been reabsorbed by one of the atoms leaving the atomic system in the entangled state. As a consequence, spontaneous decay may be regarded as a causative mechanism for inducing entanglement between the two two-level atoms when their initial state Φ is practically β | i unentangled (β 0) or to enhance it for different β in the interval π/2 β π. ≃ ≤ ≤ 5 sinβ|01〉 + cosβ|10〉 1 EOF of averaged state 0.8 EOF of initial state 0.6 F O E β 0.4 0 0.2 0 0 0.39 0.79 1.18 1.57 1.96 2.36 2.75 3.14 sinβ|00〉 + cosβ|11〉 1 0.8 0.6 F O E 0.4 β 0.2 1 0 0 0.39 0.79 1.18 1.57 1.96 2.36 2.75 3.14 β [rad] FIG. 2: The continuous lines in both plots are the EOF for mixtures of the output states occurring within the time interval where the EOF is periodic (n¯ =0). The dash-dotted lines correspond to the EOF for t=0 (initial states). In the upper plot the interval β0 < β < π/2 β0 with β0 = 1/2arcsin(1/7) demarcates the region with lower entanglement in comparison to − corresponding input states. In the bottom plot we can see that entanglement is not present in the interval ( β1,β1), with − β1 =arctan(1/8), hencedisappearing even for initially weakly entangled states. The fact that EOF exhibits strong oscillations as already illustrated may limit even its evaluation. Evaluation of the entanglement requires knowledge of the matrix elements and these cannot be determined just from one copy of anunknownquantumstate. One possibility is to havemany copiesofthe sameunknownstate andperformquantum state tomography. Recent findings show that the tomography is not necessary [17, 18] but still does hold that it is not possible to measure the entanglement from just one copy of a state. Besides, if the averagenumber of photons is so large that the time window for picking the desired state is too long comparedto the length of the oscillations, the measurement could be interpreted as a random pick from the set of all possible outputs and a proper time average must be performed. Letusstudythisquestionfortheelectromagneticfieldinitiallyinthevacuumstateasanillustration. Inthissimple case, the natural period is determined by the commeasurable Rabi frequencies √8g and √2g 2(1+sin2β) 0 0 0 1 √2π 1 0 3 sin2β 3sin2β 1 0 ̺˜(Φ)(β)= √2π ̺(β,t)d(gt)= 8 0 3s−in2β 1 3 sin−2β 0. (12) Z0 − − 0 0 0 0     The resulting EOF can be seen in the upper plot of Fig. 2 and, we are able to confirm the qualitative conclusions aboutthe entanglementbehaviorbasedonFig.1forthe averagedEOF.Asmentionedabove,wecanobservenonzero entanglement for initially factorized states. For entangled states with β < β <π/2 β the averagedEOF is lower 0 0 − thanforthe inputstates. Ingeneral,fortheremainingβ the EOFaverageishigherthanthe initialvalueofthe EOF. The extension of this averaging process for the case of an interaction with thermal light with n¯ > 0 requires the identification of a natural time scale since in this case an infinite number of the incommensurable Rabi frequencies α =2g n+1/2determinestheevolutionofthesystem. Onesuchscalecouldbefoundbythefollowingprocedure. n 1 − Writing the summations in the expression of ̺˜(Φ)(β,t) in an integral form using the Abel-Plana formula, it can be p shownthattheadimensionalparametergt 2kT/~Ωisespeciallyimportanttounderstandtheevolutionofthesystem. In fact, it can be used to define a natural time τ = g 1 ~Ω/2kT to distinguish short and long time effects of the p 0 − interaction(see Appendix for more details). If the averageprocedure from above is applied, τ could also be used for 0 p this purpose. 6 1 1 0.8 0.8 0.6 0.6 F F O O E0.4 E0.4 0.2 0.2 0 0 15 0 15 0 0.2 0.2 12 0.39 12 0.39 0.59 0.59 9 9 0.79 0.79 gt 6 0.98 β [rad] 6 0.98 β [rad] 1.18 gt 1.18 3 3 1.37 1.37 1.57 1.57 (a) (b) FIG. 3: The EOF of output state (14) for interacting atomic states Ψβ with (a) a zero photon field, (b) a thermal field with | i n¯ 0.64. Notice the small enhancement of the EOF for β π/2 where n¯ was chosen to provide a maximum value of the ≈ ∼ enhancement. B. Initial Atomic State Ψβ | i Following the same reasoning as in the previous subsection, but for an initial state of the form Ψ =sinβ 00 +cosβ 11 (13) β | i | i | i interacting with light in a thermal state (9), we arrive to a reduced density matrix with the structure b 0 0 b 1 2 0 b b 0 ̺˜(Ψ)(β,t)= 3 3 . (14) 0 b b 0 3 3 b 0 0 b  ∗2 4   The particular expressions for the matrix elements are given in Appendix. As illustrated in Fig. 3, in this case a resonant thermal electromagnetic field is not able to induce, in general, atomic entanglement for input factorized atomic states, nor to enhance it for initial already entangled states. An exception is provided for the states with β π/2. There, a very slight EOF enhancement is observed for 0 < n¯ < 2 . This effect is discussed at length in Re≃f. [3] and present, but barely observable, in Fig. 3. For the other valu∼es of β, at most, partially entangled input states exhibit revivals of the EOF. Thus, in comparison to the input state Φ , the entanglement properties of a β | i thermalfieldarenotsostriking. This qualitativeobservationis confirmedforn¯ =0 bythe mixture calculationinthe spirit of the previous section. It gives us 4(2 cos2β) 0 0 4sin2β − 1 0 cos2β cos2β 0 ̺˜(Ψ)(β)= . (15) 12 0 cos2β cos2β 0  4sin2β 0 0 6cos2β     7 0.25 0.2 0.15 1 p 0.1 0.05 0 1.57 1.37 1.18 15 0.98 12 0.79 9 0.59 0.39 6 0.2 3 β [rad] 0 gt FIG. 4: Depicted values of parameter p1 for n¯ 5.9 which determines the probability of the successful postselection of a maximally entangled state 1/√2(10 + 01 ) from≈outgoing state (14). It is possible to postselect the state irrespective of β | i | i and thusit can bedone even if the outputstate is separable. TheresultingEOFisdepictedinFig.2(bottomplot). We seethatinthe mixturesense,there isnotonlyaseparable output for initially unentangled states, as discussed above, but also for some of the input partially entangled states. Also, if we compare the EOF for the initial state (dash dotted line) and evolved state (continuous line) the output states are much less entangled in general. Nevertheless, returning to the evolved state (14), another interesting aspect appears. Eq. (14) can be written as ̺˜(Ψ)(β,t)=p ̺ +(1 p )̺ (16) 1 1 1 2 − where p b and 1 3 ≡ 1 ̺ = 01+10 01+10 1 2| ih | 1 ̺2 = b +b (b1|00ih00|+b4|11ih11|+b2|00ih11|+b∗2|11ih00|) (17) 1 4 andbothstatescanbeunambiguouslydiscriminated. Toshowthis,letusconsideraprojectivelymeasuringapparatus with three possible outputs. Two of them project into orthogonalsubspaces spanned by 00 00 and 11 11 and the | ih | | ih | third one is the complement. Then, from the form of (16) it follows that if we do not detect anything on the first and the second output the state is in the required maximally entangled state 1 01+10 . The probability of this √2| i procedure is given by the parameter p . Thus, no additional quantum resource for the discrimination is needed [19]. 1 Therefore, even if the EOF is zero (i.e. when an output state is separable) the entangled pair can be extracted in terms of postselection. Also, it is interesting that the maximum value of p is almost the same not only for different 1 input states (see Fig. 4) but also for different average photon numbers of thermal light. It is worth remarking that the higher photon averagenumber is used the less fluctuating the parameter p we get. This constantcharacterof p 1 1 is broken only for the case of a zero photon field where, not surprisingly, for an input state 00 we cannot extract a | i maximally entangled state by any means, see Eq. (15). Note that a similar procedure cannot be applied to the resulting state in Eq. (10) from the previous subsection. There, we are not able to find a decomposition of a density matrix in the form ̺ = ̺ ̺ as in the current case, 1 2 ⊕ which ensures that if two arbitrary density matrices are positive the third must be so. 8 III. CONCLUSIONS In this work, we have explored the possibility of preserving or enhancing entanglement between two atoms after their interaction with a thermal field. We have investigated the interaction for two kinds of initial atomic states Φ = 1/√2(sinβ 01 +cosβ 10 ) and Ψ = 1/√2(sinβ 00 +cosβ 11 ). Although this physical system had β β | i | i | i | i | i | i been discussed previously, new interesting phenomena have been found and the already reported results have been confirmed. For the initial state Φ , we observe a very different behavior after the interaction with the thermal β | i field for two distinguished intervals of input states. Qualitatively, for almost the whole range β =(0,π/2), the EOF is decreased while for the remaining input states there are oscillations in the EOF so that they are never smaller than their initial value. This result is more significant at low temperatures, hence, the entanglement enhancement mechanism is induced by spontaneous decay. For the secondfamily ofinput states parameterizedby Ψ , there is no entanglementenhancement for almostany β | i β. Inthis case,the entanglementis generallydegradedmatching our intuitive conceptionofaninteractionbetween a correlatedpairs of qubits and thermal light. Nevertheless, the output atomic density matrix can be decomposed into φ = 1/√2 01+10 and a partially entangled mixed state, both of them living in mutually orthogonal subspaces. | i | i Thus they can be discriminated with zero probability of error. If we consider that this holds for all β when n¯ is sufficiently high, it follows that we are able to extract the maximally entangled state φ even if the input is a | i factorized state and the output is in a separable form. We may conclude that we have at disposal a probabilistic source of maximally entangled atomic pairs, when our deposit is just a sufficiently hot thermal light with two atoms in a factorized state. Incurrentexperimentswithtrappedions,theenergiesinvolvedinthetransitionscomparedwithroomtemperatures are such that ~Ω/kT 1 for the Innsbruck group (729 nm atomic transition in 40Ca+) [2] and ~Ω/kT 1 for the NIST group (9Be+ at≪1.2 GHz) [1]. Thus, one would expect that the former is a good scenario for≫testing the predictionsmadeforthe initialstate Φ , inparticular,the entanglementenhancement. Meanwhile,theNIST group β | i parameters could be appropriate to study the possibility of stabilized postselection of the maximally entangled state described for the interaction of thermal resonant light with Ψ . β | i Acknowledgments The authors are grateful to S. Hacyan for valuable comments. Funding from CONACyT 41048-Fis acknowledged. APPENDIX Initial Atomic State Φβ | i With the help of the inverse transformation from the bare state to the dressed state basis, we rewrite the initial density matrix determined by the atomic state Φ and a thermal single-mode field β | i ̺(Φ)(β,t=0) = ∞ pn [(1+sin2β)( 3 + 4 )( 3 + 4 ) 4 −| in | in −h |n h |n n=0 X √2cos2β(( 3 + 4 ) 2 + 2 ( 3 + 4 )) 2(1 sin2β)2 2 , (A.1) − −| in | in h |n | in −h |n h |n − − | ih |n i where p = [1 exp( ~Ω/kT)]exp( n~Ω/kT). The time evolution of the whole system is directly found. Tracing n − − − over the resulting density matrix we obtain an output atomic state (Eqs. (10)) with elements 1 1 a = (1+sin2β) m h (2t) h (2t) (A.2) 1 + 2 1 8 − − 2 (cid:18) (cid:19) 1 1 1 a = (1+sin2β)(1+h (2t)) h (t)cos2β+ (1 sin2β) (A.3) 2 2 2 8 − 2 4 − 1 1 a = (1+sin2β)(1+h (2t)) (1 sin2β) (A.4) 3 2 8 − 4 − 1 1 1 a = (1+sin2β)(1+h (2t))+ h (t)cos2β+ (1 sin2β) (A.5) 4 2 2 8 2 4 − 1 1 a = (1+sin2β) m h (2t)+ h (2t) (A.6) 5 2 1 8 −− 2 (cid:18) (cid:19) 9 with ∞ pn h (t) = cosα t (A.7a) 1 n 1 n+1/2 − n=0 X ∞ 1 h (t) = p cosα t= h (t). (A.7b) 2 n n−1 −4g2 ′1′ n=0 X α = 2g n+1/2 is the Rabi frequency coming from eigenvalues of the interaction Hamiltonian (6) and m = n 1 1 −Marctan(exp( ~Ω/2kT)) where M =[1 exp( ~Ω/kT)]exp(~Ω/2kT). ± ± p − − − Expressions (A.7) appearing in the matrix elements can be written in a more compact form ∞ ∞ ∞ h (t)=2sinhκ eκ(2n+1)cos√2n+1gt=2sinh2κ eκncos√ngt e2κncos√2ngt (A.8) 2 − n=0 n=0 n=0 X X X where κ= ~Ω . Using the Abel-Plana formula 2kT ∞ ∞ 1 ∞ F(ix) F( ix) F(n)= F(x)dx+ F(0)+i − − dx (A.9) 2 e2πx 1 n=0 Z0 Z0 − X the following integral expression for h (t) is found 2 h2(t)=2sinhκ ∞e−xcost˜√xdx+4 ∞ sinxcosht˜√xπ−x sin2xcosht˜√2xdx (A.10) κ "Z0 Z0 e κ −1 # with an adimensional variable t˜ = gt that establishes a natural time scale τ = √κ for the description of the √κ 0 g interaction. Notice that the second integral is an exponential decreasing function of the temperature so that for a sufficiently hot environment h (t) 2sinhκ ∞e xcost˜√xdx=2sinhκ 1 t˜e t˜2 t˜ex2dx . (A.11) 2 − − ∼ κ Z0 κ " − Z0 # The latter integral can be recognized as an error function with an imaginary argument. An integral expression for h (t) can be found in a similar manner. 1 Initial Atomic State Ψβ | i Calculations similar to the case of the initial state Φ lead to the following matrix coefficients of Eq. 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Lett. 78, 2275 [13] Plenio M B and Virmani S,quant-ph/0504163 [14] Z˙yczkowski K, Horodecki P, Sanpera A, and Lewenstein M 1998 Phys. Rev. A 58, 883; Vidal G and Werner R F 2002 Phys. Rev. A 65, 032314 [15] Popescu SandRohrlichD1997 Phys. Rev. A56, R3319; Donald MJ, HorodeckiM,and RudolphO2002 J. Math. Phys. 43, 4252 [16] Lee S,Chi D P, Oh S D,and Kim J 2003 Phys. Rev. A 68, 062304 [17] Ac´ın A,Tarrach R,and Vidal G 2000 Phys. Rev. A 61, 062307 [18] HorodeckiP 2003 Phys. Rev. Lett. 90, 167901 [19] If, from some reasons, we need to preserve both output states ̺1,̺2 intact we introduce an auxiliary qubit |0iaux and the unitary matrix T = CNOT13CNOT23 where in CNOTmn m and n is a control and a target qubit, respectively. T Thisis usefulfor non-demolition discrimination between̺1 and ̺2 since underthistransformation we get ̺˜ 0 0aux ⊗| ih | → p1̺1 1 1aux+(1 p1)̺2 0 0aux.Sothat,byaprojectivemeasurementontheancilla,̺1and̺2canbeunambigously ⊗| ih | − ⊗| ih | discriminated without destroying them.

Entanglement enhancement and postselection for two atoms interacting with thermal light

Similar Entanglement enhancement and postselection for two atoms interacting with thermal light